What is scientific notation?

Scientific notation expresses any number as a coefficient (a) multiplied by a power of 10: a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. For example, 93,000,000 becomes 9.3 × 10⁷ and 0.000456 becomes 4.56 × 10⁻⁴. This format makes very large and very small numbers compact and easier to compare. It is standard in physics, chemistry, astronomy, and engineering.

How do you write a number in scientific notation?

Move the decimal point until the number is between 1 and 10. The number of places you moved — and in which direction — becomes the exponent. Moving left (for numbers ≥ 10) gives a positive exponent; moving right (for numbers < 1) gives a negative exponent. For example, 47,000: move the decimal 4 places left → 4.7 × 10⁴. For 0.0023: move the decimal 3 places right → 2.3 × 10⁻³.

How do you multiply numbers in scientific notation?

Multiply the coefficients and add the exponents: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10^(m+n). Then normalize so the coefficient is between 1 and 10. For example: (3.0 × 10⁸) × (2.5 × 10⁴) = 7.5 × 10¹². For division, divide the coefficients and subtract the exponents: (6.0 × 10⁸) ÷ (2.0 × 10³) = 3.0 × 10⁵.

E notation is a calculator and computer shorthand for scientific notation where "E" replaces "× 10^". For example, 4.56 × 10⁻⁴ is written as 4.56e-4 or 4.56E-4. Most programming languages and spreadsheets use this format for very large or very small floating-point numbers. When you see "1.23e10" in code or a spreadsheet, it means 1.23 × 10¹⁰ = 12,300,000,000.

How do you add numbers in scientific notation?

To add or subtract numbers in scientific notation, first convert them to the same exponent, then add or subtract the coefficients, then normalize. For example: 3.5 × 10⁴ + 2.1 × 10³ = 3.5 × 10⁴ + 0.21 × 10⁴ = 3.71 × 10⁴. If the exponents differ greatly, the smaller number contributes negligibly. This calculator handles the conversion automatically — just enter your numbers and select the operation.

When is scientific notation used?

Scientific notation is essential when working with numbers that have many zeros — either very large or very small. Common examples: the speed of light is 3.0 × 10⁸ m/s; the mass of an electron is 9.1 × 10⁻³¹ kg; the distance to the nearest star (Proxima Centauri) is 4.0 × 10¹⁶ m; a human cell is about 1.0 × 10⁻⁵ m in diameter. In finance, national debt figures and market capitalizations also benefit from scientific notation for comparison.

Why do scientists use scientific notation instead of writing out all the zeros?

Writing 602,214,076,000,000,000,000,000 (Avogadro's number) is error-prone — it is easy to miscounts zeros. In scientific notation it is 6.022 × 10²³, which immediately communicates both the magnitude and the precision. Scientists also use scientific notation to track significant figures: 3.0 × 10⁸ m/s (2 sig figs) is clearly less precise than 2.998 × 10⁸ m/s (4 sig figs), while "300,000,000" is ambiguous about precision.

What are some famous constants in scientific notation?

Key constants worth knowing: speed of light c = 3.0 × 10⁸ m/s; Avogadro's number Nₐ = 6.022 × 10²³ mol⁻¹; electron charge e = 1.602 × 10⁻¹⁹ C; Planck's constant h = 6.626 × 10⁻³⁴ J·s; Earth's mass = 5.97 × 10²⁴ kg; mass of a proton = 1.67 × 10⁻²⁷ kg; diameter of a hydrogen atom ≈ 1.06 × 10⁻¹⁰ m. These span 55 orders of magnitude — exactly why scientific notation exists.

Scientific Notation Calculator — Convert & Calculate

Converts numbers to and from scientific notation and performs arithmetic, showing results in scientific notation, E notation, and standard form.

How to Use the Scientific Notation Calculator

This scientific notation calculator converts numbers and performs arithmetic in scientific notation. Select Convert Number to convert any decimal number into scientific notation, E notation, and standard form. Or switch to Arithmetic mode to add, subtract, multiply, or divide two numbers expressed as coefficient × 10ⁿ — results appear in all three formats automatically.

In convert mode, type the number in standard decimal form (e.g., 0.000456 or 93000000). In arithmetic mode, enter each number as a coefficient and an exponent separately. For numbers involving significant digits, see our rounding calculator.

Scientific Notation Rules

Scientific notation follows a strict format: a × 10ⁿwhere the coefficient a satisfies 1 ≤ |a| < 10 and n is an integer (positive, negative, or zero).

Large numbers — positive exponent: 5,280 = 5.280 × 10³
Small numbers — negative exponent: 0.00012 = 1.2 × 10⁻⁴
Numbers between 1 and 10 — exponent is 0: 7.3 = 7.3 × 10⁰
Negative numbers — the negative sign is part of the coefficient, not the exponent: −4.5 × 10² = −450

E Notation

E notation replaces "× 10^" with the letter E. It is used in calculators, programming languages (Python, JavaScript, C), and spreadsheets (Excel, Google Sheets). Examples: 6.022e23 (Avogadro's number), 1.602e-19 (electron charge in coulombs), 2.998e8 (speed of light in m/s).

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Arithmetic in Scientific Notation

Multiplication and Division

Multiplication: multiply coefficients, add exponents. (3.0 × 10⁸) × (2.5 × 10⁴) = 7.5 × 10¹².

Division: divide coefficients, subtract exponents. (6.0 × 10⁸) ÷ (3.0 × 10³) = 2.0 × 10⁵.

Addition and Subtraction

These are more involved because you must first convert to a common exponent: (3.5 × 10⁴) + (2.1 × 10³) = (3.5 × 10⁴) + (0.21 × 10⁴) = 3.71 × 10⁴. The calculator handles this conversion automatically. For large differences in magnitude, the smaller number may be negligible.

Significant Figures in Scientific Notation

Scientific notation makes counting significant figures straightforward. The number of digits in the coefficient equals the number of significant figures: 4.50 × 10³ has 3 significant figures; 4.5 × 10³ has 2. Trailing zeros after the decimal point are significant. Zeros in scientific notation are never ambiguous — 4500 has 2–4 sig figs depending on context, but 4.500 × 10³ unambiguously has 4.

For detailed significant figure calculations, our decimal calculator supports operations on decimal numbers with precision control.

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Real-World Scientific Notation Values

Speed of light: 2.998 × 10⁸ m/s
Avogadro's number: 6.022 × 10²³ mol⁻¹
Electron mass: 9.109 × 10⁻³¹ kg
Earth–Sun distance: 1.496 × 10¹¹ m
Human cell diameter: ~1.0 × 10⁻⁵ m
US national debt (approx.): ~3.5 × 10¹³ USD
Hydrogen atom diameter: ~1.06 × 10⁻¹⁰ m

Sources & References

Scientific notation — Wikipedia
Significant figures — Wikipedia

Scientific Notation Calculator